reserve A, B, X, Y for set;

theorem Th5:
  <:pr2(X,Y),pr1(X,Y):>.:[:A,B:] c= [:B,A:]
proof
  set f = <:pr2(X,Y),pr1(X,Y):>;
  let y be object;
  assume y in f.:[:A,B:];
  then consider x being object such that
A1: x in dom f and
A2: x in [:A,B:] and
A3: y = f.x by FUNCT_1:def 6;
  consider x1, x2 being object such that
A4: x1 in A & x2 in B and
A5: x = [x1,x2] by A2,ZFMISC_1:def 2;
  dom f = [:X,Y:] by Th4;
  then x1 in X & x2 in Y by A1,A5,ZFMISC_1:87;
  then f.(x1,x2) = [x2,x1] by Lm1;
  hence thesis by A3,A4,A5,ZFMISC_1:87;
end;
